Question
Raoul gave two musical performances. His second performance occurred 86 days after his first performance. If Raoul’s second performance was on a Tuesday, on what day of the week was his first performance?
- ASunday
- BMonday
- CWednesday
- DThursday
Show the answer
Correct answer: A
The total number of days (86) is divided by the number of days in a week (7).
The 2 days are counted back from Tuesday, so the first performance was on a Sunday.
Question analysis
Raoul's second musical performance came 86 days after his first and fell on a Tuesday. Students work out the day of the week of the first performance. It tests division with remainders in a repeating weekly cycle, and the traps are counting in the wrong direction or miscounting the leftover days.
- NYS standards
- 6.NS.2 multi-digit division
- Grade level
- Grade 6 content, by NYS standard
- Compared with grade 7
- At grade 7 Dividing with a remainder is grade 6 arithmetic, but using the remainder to step through a seven-day cycle, and counting backward, is less routine than grade 7 classwork. It needs care with direction rather than new content.
- Estimated difficulty
- Moderate: several steps, or a close reading with one strong distractor.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly apply
- What makes it hard
- No obvious method moderate
- Turning words into math minor
- Solution
- 2 steps
- Reading level
- Question and choices: about grade 5, 36 words.
- Students, last 30 days
- Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)
Why it is hard
Counting 86 days one at a time is hopeless in the time allowed, so the question rewards seeing that whole weeks return to the same weekday. Two decisions remain after that, and each has a listed wrong answer: which way to count, and whether Tuesday itself is one of the 2 days. Saying the days aloud, Monday is 1 back and Sunday is 2 back, avoids both.
No obvious method moderate Working it out
Nothing in the stem says to divide by 7; the key is that 84 of the 86 days are exactly 12 weeks, which leaves only 2 days to count. (Schoenfeld, 1985)
Turning words into math minor Understanding the question
"His second performance occurred 86 days after his first" puts the unknown day before the Tuesday, so the leftover days are counted backward. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
Strategy: Remove whole weeks from the 86 days, since each full week lands on the same weekday, then count the leftover days backward from Tuesday.
Traps
- Negatives and direction: Arithmetic with negative numbers, the order of a subtraction, or direction on a number line. (Moving a term across the equals sign without changing its sign is a faulty procedure.)
- Counting the ends: Inclusive or exclusive ends, fence posts, consecutive numbers: an answer that is one too many or too few, or an endpoint drawn open where it belongs closed (or the reverse).
The wrong answers, classified
B Counting the ends (Brown & Burton, 1978)
Counts Tuesday itself as the first of the 2 leftover days, so moves back only one day, to Monday.
C Negatives and direction (Brown & Burton, 1978)
Counts forward from Tuesday instead of back and also counts Tuesday as one of the 2 days, landing on Wednesday.
D Negatives and direction (Brown & Burton, 1978)
Counts the 2 leftover days forward from Tuesday, finding the day 86 days after the second performance instead of before it.
Worked solution, step by step
Formulas used
- Weeks and leftover days
- days is the total number of days. weeks is the number of full weeks inside it. leftover is the days that remain, from 0 to 6. The day of the week repeats every 7 days, so only the leftover days change it.
Sort out the question
The second show was on a Tuesday. It came 86 days after the first show. So the first show was 86 days before that Tuesday.
You need the day of the week of the first show. So you count back 86 days from Tuesday.
Take out the full weeks
Counting back 86 days one at a time is slow. Here is a shortcut. Every 7 days, the same weekday comes back. So divide 86 by 7.
That is 12 full weeks and 2 extra days. Going back 12 weeks from a Tuesday lands on a Tuesday again.
Count back the 2 extra days
Start on Tuesday and go back one day at a time. One day back is Monday. Two days back is Sunday.
Watch out: Do not count Tuesday itself as one of the 2 days. That lands on Monday, choice B. And count backward, not forward. Counting forward 2 days gives Thursday, choice D.
So the answer is A, Sunday.
Check your answer
Count forward to be sure. Start on a Sunday. After 84 days, which is 12 weeks, it is Sunday again. Two more days are Monday and Tuesday. That is day 86, a Tuesday. It matches.
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